[FOM] SOL vs. ZFC

Quoting Joe Shipman <JoeShipman at aol.com>:
> 1) is there any commonly used system of axioms for second order
> logic that includes sentences which ZFC does not prove are validities?
Surely the soundness of all commonly used systems of axioms for
second-order logic is provable in ZFC?
> 2) is there any sentence which ZFC proves is a validity of second
> order logic, but which is not a consequence of any commonly used
> system of axioms for second order logic other than the system
> "enumerate sentences with ZFC-proofs of their validity"?
Surely that's not a "commonly used system of axioms for second
order logic". In any case, again I'd be very surprised if this didn't
hold for all commonly used axioms for second order logic. Given any
commonly used set of axioms and rules of inference for second-order
logic, provided they're provably (in ZFC) sound, it's provable in ZFC
that "second-order arithmetic implies the consistency of the deductive
closure of second-order arithmetic under the axioms and rules" is a
logically valid.
--
Aatu Koskensilta (aatu.koskensilta at uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus